We only know that he screams out in emotional pain after he snaps Zod's neck. We do not know why. I would feel quite different about this had the movie built up a "never kill" policy that he was forced to violate, or even something simple like the exchange with Lois from Alan Moore's Whatever Happened to the man of Tomorrow story. But we didn't get that...they had to cut the emotional scene short so Superman could crack jokes.

Maybe it's just me, but I thought Superman clearly screamed because he was forced to kill Zod. It was so straightforward. Clark's hesitation to fight the bully. Clark hesitating to fight the man at the bar. Zod saying he will stop at nothing to kill every human on the planet. The extra emphasis on a family about to be slaughtered by Zod's heat vision. Superman's hesitation as Zod's heat vision got closer and closer to the victims. Like I said, it was straightforward.

Given the context of all of that, Superman screaming because he killed the last remaining Kryptonian is reaching.